Roster of Union Soldiers, 1861-1865, shows:

Baker, Ira R. WI 29th Inf. Co.E

Organized at Camp Randall, Racine, WI 11/27/1862
At a point on the East bank of the Mississippi River, opposite Helena, AR, pulled picket duty to 12/23/1862
Moved to Frier's Point and encamped
01/11/1863, went to Devall's Bluff, AR where it captured artillery, arms, stores & prisoners
Feb 1863, Yazoo Pass expedition
March 1863, assigned to 13th Corps & sent to Vicksburg
Fought at Port Gibson
Engaged at Champion's Hill
Siege of Vicksburg
July 1863, 2nd battle of Jackson
Guard & Picket duty during summer & fall, 1863
Texas Expedition; Carrion Crow bayou; Cavallo Pass; Cloutierville; Sabine Cross Roads
July 1864, Algiers; 1st Brigade, Provisional Division
Skirmish Atchafalaya River
Clinton, LA; 2nd Brigade, 2nd Div, 19th Corps; Morganza via Port Hudson
09/03/1864, St Charles, AR - guard duty to 10/23/1864
Remainder of year spent in heavy fatigue & picket duty
Jan 1865, Kennersville near New Orleans to 02/05/1865
move against Spanish Fort to 03/31/1865
Ordered to Fort Blakely & provost Guard at Mobile to 05/26/1865; then to Shreveport & same duty
Mustered out 06/29/1865

The Ira Baker shown above is the only Ira Baker from Wisconsin I can find.

===========================================================
Baker's serving in the 23rd WI Inf Regt were:

Baker, Alpheus W. WI 23rd Inf. Co.H 2nd Lt.
Baker, Andrew WI 23rd Inf. Co.K Cpl.
Baker, Daniel WI 23rd Inf. Co.K
Baker, James L. WI 23rd Inf. Co.D 1st Lt.
Baker, William H. WI 23rd Inf. Co.D Sgt.

Organized at Camp Randall, Madison, WI in Aug 1862
09/15/1862 to Cincinnati, then South to Vicksburg
Assault on Chicasaw Bluffs & reduction of Arkansaw Post
To young's Point, LA, near Vicksburg [3/4 of men stricken with severe virulent camp diseases]
On scout & foraging duty to 04/30/1863
Reserve at Port Gibson
Champion's Hill; Black River Bridge [captured 60th TN]
Reached Vicksburg on 18th; general assault on 22nd; on duty to surrender [150 men fit for duty]
Battle of Jackson & on duty to evacuation
Expedition through LA, occupying Opelousas entire summer
Carrion Crow Bayou in Nov 1863
Brashear City 12/13/1863
Ordered to Texas where it remained until 02/22/1864; returned to LA
Red River Expedition; Sabine Cross Roads; Cloutierville
Encamped Baton Rouge, 05/25/1864 to 07/08/1864; moved to Algiers & Morganza to 08/18/1864
Transferred to 3rd Brigade, 2nd Div, 19th Corps 7 engaged in Guard, Post, Garrison & recon to 05/01/1865
Ordered to Mobile; engaged in Siege, Patrol & Picket duty to 07/04/1865 when it mustered out
[The Union Army, Vol IV]